Differential Sheaves And Connections: A Natural Approach To Physical Geometry

Prolegomena: Exordium; Basic Working Notions; Observables and States; Connections and Differential Analysis; The Functorial Imperative; Grothendieck Topos Interpretation of the Hom-Tensor Adjunction; The Grothendieck Topology of Epimorphic Families; Unit and Counit of the Hom-Tensor Adjunction; General Theory: General Introduction; Basic Assumptions of ADG (Abstract Differential Geometry); Basic Framework; Bohr's Correspondence; Functorial, Topos-Theoretic Mechanism of ADG; Kahler Construction; Elementary Particles in the Jargon of ADG; Relational Aspect of Space, Again; Dynamical Dressing, Extension: Kahler Construction (Contn'd); Adjunction, Least Action Principle; Transformation Law of Potentials, in Terms of ADG; Characteristics of a Physical Law; Complementary Remarks; Epilogue; Applications: Fundamental Adjunctions: On Utiyama's Theme/Principle Through "A-Invariance"; "Affine Geometry" and "Quantum"; Chasing Feynman; Stone-von Neumann Adjunction; Quantized Einstein's Equation; The Essence of ADG; Peroration;